Abstract

The aim of this paper is to present new results related to the convergence of the sequence of the 𝑞-Bernstein polynomials {𝐵𝑛,𝑞(𝑓;𝑥)} in the case 𝑞>1, where 𝑓 is a continuous function on [0,1]. It is shown that the polynomials converge to 𝑓 uniformly on the time scale 𝕁𝑞={𝑞−𝑗}∞𝑗=0∪{0}, and that this result is sharp in the sense that the sequence {𝐵𝑛,𝑞(𝑓;𝑥)}∞𝑛=1 may be divergent for all 𝑥∈𝑅⧵𝕁𝑞. Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.

1. Introduction

Let 𝑓∶[0,1]→ℂ, 𝑞>0, and 𝑛∈ℕ. Then, the q-Bernstein polynomial of 𝑓 is defined by
𝐵𝑛,𝑞(𝑓;𝑥)=𝑛𝑘=0𝑓[𝑘]𝑞[𝑛]𝑞𝑝𝑛𝑘(𝑞;𝑥),(1.1)
where
𝑝𝑛𝑘⎡⎢⎢⎣𝑛𝑘⎤⎥⎥⎦(𝑞;𝑥)=𝑞𝑥𝑘(𝑥;𝑞)𝑛−𝑘,𝑘=0,1,…𝑛,(1.2)
with 𝑛𝑘𝑞 being the q-binomial coefficients given by
⎡⎢⎢⎣𝑛𝑘⎤⎥⎥⎦𝑞=[𝑛]𝑞![𝑘]𝑞![]𝑛−𝑘𝑞!,(1.3)
and (𝑥;𝑞)𝑚 being the 𝑞-Pochhammer symbol:
(𝑥;𝑞)0=1,(𝑥;𝑞)𝑚=𝑚−1𝑠=0(1−𝑥𝑞𝑠),(𝑥;𝑞)∞=∞𝑠=0(1−𝑥𝑞𝑠).(1.4)
Here, for any nonnegative integer 𝑘,
[𝑘]𝑞[1]!=𝑞[2]𝑞⋯[𝑘]𝑞[0](𝑘=1,2,…),𝑞!∶=1(1.5)
are the q-factorials with [𝑘]𝑞 being the q-integer given by
[𝑘]𝑞=1+𝑞+⋯+𝑞𝑘−1[0](𝑘=1,2,…),𝑞∶=0.(1.6)
We use the notation from [[1], Ch. 10].

The polynomials 𝑝𝑛0(𝑞;𝑥),𝑝𝑛1(𝑞;𝑥),…,𝑝𝑛𝑛(𝑞;𝑥), called the 𝑞-Bernstein basic polynomials, form the 𝑞-Bernstein basis in the linear space of polynomials of degree at most 𝑛.

Although, for 𝑞=1, the 𝑞-Bernstein polynomial 𝐵𝑛,𝑞(𝑓;𝑥) turns into the classical Bernstein polynomial 𝐵𝑛(𝑓;𝑥):
𝐵𝑛(𝑓;𝑥)=𝑛𝑘=0𝑓𝑘𝑛⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠𝑥𝑘(1−𝑥)𝑛−𝑘,(1.7)
conventionally, the name “𝑞-Bernstein polynomials” is reserved for the case 𝑞≠1.

Based on the 𝑞-Bernstein polynomials, the 𝑞-Bernstein operator on 𝐶[0,1] is given by
𝐵𝑛,𝑞∶𝑓↦𝐵𝑛,𝑞(𝑓;⋅).(1.8)
A detailed review of the results on the 𝑞-Bernstein polynomials along with an extensive bibliography has been provided in [2]. In this field, new results concerning the properties of the 𝑞-Bernstein polynomials and/or their various generalizations are still coming out (see, e.g, papers [3–8], all of which have appeared after [2]).

The popularity of the 𝑞-Bernstein polynomials is attributed to the fact that they are closely related to the 𝑞-binomial and the 𝑞-deformed Poisson probability distributions (cf. [9]). The 𝑞-binomial distribution plays an important role in the 𝑞-boson theory, providing a 𝑞-deformation for the quantum harmonic formalism. More specifically, it has been used to construct the binomial state for the 𝑞-boson. Meanwhile, the 𝑞-deformed Poisson distribution, which is the limit form of 𝑞-binomial one, defines the energy distribution in a 𝑞-analogue of the coherent state [10]. Another motivation for this study is that various estimates related to the natural sequences of functions and operators in functional spaces, convergence theorems, and estimates for the rates of convergence are of decisive nature in the modern functional analysis and its applications (see, e.g., [4, 11, 12]).

The 𝑞-Bernstein polynomials retain some of the properties of the classical Bernstein polynomials. For example, they possess the end-point interpolation property:
𝐵𝑛,𝑞(𝑓;0)=𝑓(0),𝐵𝑛,𝑞(𝑓;1)=𝑓(1),𝑛=1,2,…,𝑞>0,(1.9)
and leave the linear functions invariant:
𝐵𝑛,𝑞(𝑎𝑡+𝑏;𝑥)=𝑎𝑥+𝑏,𝑛=1,2,…,𝑞>0.(1.10)
In addition, the 𝑞-Bernstein basic polynomials (1.2) satisfy the identity
𝑛𝑘=0𝑝𝑛𝑘(𝑞;𝑥)=1∀𝑛=1,2,…,∀𝑞>0.(1.11)
Furthermore, the 𝑞-Bernstein polynomials admit a representation via the divided differences given by (3.3), as well as demonstrate the saturation phenomenon (see [2, 7, 13]).

Despite the similarities such as those indicated above, the convergence properties of the 𝑞-Bernstein polynomials for 𝑞≠1 are essentially different from those of the classical ones. What is more, the cases 0<𝑞<1 and 𝑞>1 in terms of convergence are not similar to each other, as shown in [14, 15]. This absence of similarity is brought about by the fact that, for 0<𝑞<1, 𝐵𝑛,𝑞 are positive linear operators on 𝐶[0,1], whereas for 𝑞>1, no positivity occurs. In addition, the case 𝑞>1 is aggravated by the rather irregular behavior of basic polynomials (1.2), which, in this case, combine the fast increase in magnitude with the sign oscillations. For a detailed examination of this situation, see [16], where, in particular, it has been shown that the norm ‖𝐵𝑛,𝑞‖ increases rather rapidly in both 𝑛 and 𝑞. Namely,
‖‖𝐵𝑛,𝑞‖‖∼2𝑒⋅𝑞𝑛(𝑛−1)/2𝑛as𝑛→∞,𝑞→+∞.(1.12)
This puts serious obstacles in the analysis of the convergence for 𝑞>1. The challenge has inspired some papers by a number of authors dealing with the convergence of 𝑞-Bernstein polynomials in the case 𝑞>1 (see, e.g., [7, 17]). However, there are still many open problems related to the behavior of the 𝑞-Bernstein polynomials with 𝑞>1 (see the list of open problems in [2]).

In this paper, it is shown that the time scale
𝕁𝑞=𝑞−𝑗∞𝑗=0∪{0}(1.13)
is the “minimal” set of convergence for the 𝑞-Bernstein polynomials of continuous functions with 𝑞>1, in the sense that every sequence {𝐵𝑛,𝑞(𝑓;𝑥)} converges uniformly on 𝕁𝑞. Moreover, it is proved that 𝕁𝑞 is the only set of convergence for some continuous functions.

The paper is organized as follows. In Section 2, we present results concerning the convergence of the 𝑞-Bernstein polynomials on the time scale 𝕁𝑞. Section 3 is devoted to the 𝑞-Bernstein polynomials of the Weierstrass-type functions. Some of the results throughout the paper are also illustrated using numerical examples.

2. The Convergence of the 𝑞-Bernstein Polynomials on 𝕁𝑞

In this paper, 𝑞>1 is considered fixed. It has been shown in [15], that, if a function 𝑓 is analytic in 𝐷𝜀={𝑧∶|𝑧|<1+𝜀}, then it is uniformly approximated by its 𝑞-Bernstein polynomials on any compact set in 𝐷𝜀, and, in particular, on [0,1].

In this study, attention is focused on the 𝑞-Bernstein polynomials of “bad” functions, that is, functions which do not have an analytic continuation from [0,1] to the unit disc. In general, such functions are not approximated by their 𝑞-Bernstein polynomials on [0,1]. Moreover, their 𝑞-Bernstein polynomials may tend to infinity at some points of [0,1] (a simple example has been provided in [15]). Here, it is proved that the divergence of {𝐵𝑛,𝑞(𝑓;𝑥)} may occur everywhere outside of 𝕁𝑞, which is a “minimal” set of convergence.

However, in spite of this negative information, it will be shown that, for any 𝑓∈𝐶[0,1], the sequence of its 𝑞-Bernstein polynomials converges uniformly on the time scale 𝕁𝑞.

The next statement generalizing Lemma 1 of [15] can be regarded as a discrete analogue of the Popoviciu Theorem.

Theorem 2.1. Let 𝑓∈𝐶[0,1]. Then
||𝐵𝑛,𝑞𝑓;𝑞−𝑗𝑞−𝑓−𝑗||≤2𝜔𝑓⎛⎜⎜⎝𝑞−𝑗1−𝑞−𝑗[𝑛]𝑞⎞⎟⎟⎠,𝑗∈ℤ+,(2.1)
where 𝜔𝑓 is the modulus of continuity of 𝑓 on [0,1].

Corollary 2.2. If 𝑗∈ℤ+, then
||𝐵𝑛,𝑞𝑓;𝑞−𝑗𝑞−𝑓−𝑗||≤2𝜔𝑓12√[𝑛]𝑞,(2.2)
that is, 𝐵𝑛,𝑞(𝑓;𝑥) converges uniformly to 𝑓(𝑥) on the time scale 𝕁𝑞.

Proof. The proof is rather straightforward. First, notice that 𝑝𝑛𝑘(𝑞;𝑞−𝑗)≥0 for all 𝑛,𝑘,𝑗, while ∑𝑛𝑘=0𝑝𝑛𝑘(𝑞;𝑞−𝑗)=1 by virtue of (1.11). Then
||𝐵𝑛,𝑞𝑓;𝑞−𝑗𝑞−𝑓−𝑗||≤𝑛𝑘=0||||𝑓[𝑘]𝑞[𝑛]𝑞𝑞−𝑓−𝑗||||𝑝𝑛𝑘𝑞;𝑞−𝑗≤𝑛𝑘=0𝜔𝑓||||[𝑘]𝑞[𝑛]𝑞−𝑞−𝑗||||𝑝𝑛𝑘𝑞;𝑞−𝑗≤𝜔𝑓(𝛿)𝑛𝑘=011+𝛿2[𝑘]𝑞[𝑛]𝑞−𝑞−𝑗2𝑝𝑛𝑘𝑞;𝑞−𝑗(2.3)
for any 𝛿>0. Plain calculations (see, e.g., [13], formula (2.7)) show that
𝐵𝑛,𝑞(𝑡−𝑥)2=;𝑥𝑥(1−𝑥)[𝑛]𝑞,(2.4)
which implies that
||𝐵𝑛,𝑞𝑓;𝑞−𝑗𝑞−𝑓−𝑗||≤𝜔𝑓1(𝛿)⋅1+𝛿2⋅𝑞−𝑗1−𝑞−𝑗[𝑛]𝑞.(2.5)
Then, one can immediately derive the result by choosing √𝛿=𝑞−𝑗(1−𝑞−𝑗)/[𝑛]𝑞.

Remark 2.3. In [7], Wu has shown that if 𝑓∈𝐶1[0,1], then for any 𝑗∈ℤ+, one has:
||𝐵𝑛,𝑞𝑓;𝑞−𝑗𝑞−𝑓−𝑗||≤𝐶𝑗(𝑞−𝑛),𝑛→∞,where𝐶𝑗→∞as𝑗→∞.(2.6)
The condition 𝑓∈𝐶1[0,1] cannot be left out completely, as the following example shows.

Example 2.4. Consider a function 𝑓∈𝐶[0,1] satisfying
⎧⎪⎨⎪⎩0𝑓(𝑥)=if𝑥∈0,𝑞−2∪𝑞−1,𝑞,1−1−𝑥𝛼if𝑞𝑥∈−2+𝑞−12,𝑞−1,(2.7)
where 0<𝛼<1. Then, for 𝑛 large enough, we have
𝐵𝑛,𝑞𝑓;𝑞−1𝑞−𝑓−1[]=𝑓𝑛−1𝑞[𝑛]𝑞𝑝𝑛,𝑛−1𝑞;𝑞−1=𝑞−1𝑞𝛼⋅(𝑞𝑛−1)1−𝛼𝑞𝑛≥𝐶𝑞−𝑛𝛼,(2.8)
where 𝐶 is a positive constant independent from 𝑛.

As it has been already mentioned, the behavior of the 𝑞-Bernstein polynomials in the case 𝑞>1 outside of the time scale 𝕁𝑞 may be rather unpredictable. The next theorem shows that the sequence {𝐵𝑛,𝑞(𝑓;𝑥)} may be divergent for all 𝑥∈ℝ⧵𝕁𝑞.

Proof. The 𝑞-Bernstein polynomial of 𝑓 is
𝐵𝑛,𝑞(𝑓;𝑥)=𝑛𝑘=0[𝑘]𝑞[𝑛]𝑞𝛼𝑝𝑛𝑘1(𝑞;𝑥)=[𝑛]𝛼𝑞𝑛𝑘=1[𝑘]𝛼𝑞𝑝𝑛𝑘(𝑞;𝑥).(2.10)
Since for 𝑘=1,2,…,𝑛−1 one has
𝑝𝑛𝑘(𝑞;𝑥)=(𝑞𝑛𝑞−1)⋯𝑛−𝑘+1−1𝑞𝑘𝑥−1⋯(𝑞−1)𝑘(𝑥;𝑞)𝑛−𝑘=𝑞(2𝑛−𝑘+1)𝑘/2(𝑞−𝑛;𝑞)𝑘𝑞𝑘−1⋯(𝑞−1)(−1)𝑛−𝑘𝑞(𝑛−𝑘)(𝑛−𝑘−1)/2𝑥𝑛1𝑥;1𝑞𝑛−𝑘=(−1)𝑛𝑞𝑛(𝑛−1)/2𝑥𝑛⋅(−1)𝑘𝑞𝑘(𝑞−𝑛;𝑞)𝑘𝑞𝑘⋅1−1⋯(𝑞−1)𝑥;1𝑞𝑛−𝑘,(2.11)
it follows that
𝐵𝑛,𝑞(𝑓;𝑥)=(−1)𝑛𝑞𝑛(𝑛−1)/2𝑥𝑛[𝑛]𝛼𝑞⋅𝑇(𝑛,𝑞,𝑥),(2.12)
where
𝑇(𝑛;𝑞;𝑥)∶=(−1)𝑛[𝑛]𝛼𝑞𝑞𝑛(𝑛−1)/2+𝑛−1𝑘=0(−1)𝑘[𝑘]𝛼𝑞𝑞𝑘(𝑞−𝑛;𝑞)𝑘𝑞𝑘⋅1−1⋯(𝑞−1)𝑥;1𝑞𝑛−𝑘.(2.13)
Obviously,
lim𝑛→∞𝑞𝑛(𝑛−1)/2𝑥𝑛[𝑛]𝛼𝑞=∞forany𝑥≠0.(2.14)
As such, the theorem will be proved if it is shown that
lim𝑛→∞𝑇(𝑛,𝑞,𝑥)≠0for𝑥∉𝕁𝑞.(2.15)
As lim𝑛→∞(𝑞−𝑛(𝑛−1)/2(−1)𝑛[𝑛]𝛼𝑞)=0, it suffices to prove that
lim∞𝑛→∞𝑘=0𝑐𝑘𝑛≠0when𝑥∉𝕁𝑞,(2.16)
where
𝑐𝑘𝑛⎧⎪⎨⎪⎩∶=(−1)𝑘[𝑘]𝛼𝑞𝑞𝑘(𝑞−𝑛;𝑞)𝑘𝑞𝑘⋅1−1⋯(𝑞−1)𝑥;1𝑞𝑛−𝑘if0𝑘≤𝑛−1,if𝑘≥𝑛.(2.17)
The fact that (𝑞−𝑛;𝑞)𝑘≤1 and the inequality
||||1𝑥;1𝑞𝑛−𝑘||||≤−1;1|𝑥|𝑞𝑛−𝑘≤−1;1|𝑥|𝑞∞(2.18)
lead to
||𝑐𝑘𝑛||≤𝑞𝑘[𝑘]𝛼𝑞𝑞𝑘−1−1⋯(𝑞−1);1|𝑥|𝑞∞=∶𝑑𝑘.(2.19)
Now, since
lim𝑛→∞𝑐𝑘𝑛=(−1)𝑘𝑞𝑘[𝑘]𝛼𝑞𝑞𝑘1−1⋯(𝑞−1)𝑥;1𝑞∞,(2.20)
and the series ∑∞𝑘=0𝑑𝑘 is convergent, the Lebesgues dominated convergence theorem implies
lim∞𝑛→∞𝑘=0𝑐𝑘𝑛=∞𝑘=0lim𝑛→∞𝑐𝑘𝑛=1(𝑞−1)𝛼1𝑥;1𝑞∞⋅∞𝑘=0(−1)𝑘𝑎𝑘,(2.21)
where 𝑎𝑘=𝑞𝑘(𝑞𝑘−1)𝛼/(𝑞𝑘−1)⋯(𝑞−1),𝑘=1,2,…. Moreover,
1(𝑞−1)𝛼1𝑥;1𝑞∞≠0whenever𝑥∉𝕁𝑞.(2.22)
How about the sum of the series in (2.21)? Consider the following two cases. Case 1. 0<𝛼<1/3.Let us show that 𝑎𝑘+1<𝑎𝑘, 𝑘=1,2,… for 𝑞≥2. Since
𝑎𝑘+1𝑎𝑘=𝑞𝑞𝑘+1−11−𝛼𝑞𝑘−1𝛼,(2.23)
for 𝑘≥2 it follows that
𝑎𝑘+1𝑎𝑘≤𝑞𝑞3−11−𝛼𝑞2−1𝛼≤𝑞𝑞(𝑞−1)2+𝑞+11−𝛼(𝑞+1)𝛼≤𝑞(𝑞−1)(𝑞+1)<1.(2.24)
Notice that (2.24) holds for any 𝛼∈(0,1). In addition, if 𝑘=1, then
𝑎2𝑎1=𝑞𝑞2−11−𝛼(𝑞−1)𝛼=𝑞(𝑞−1)(𝑞+1)1−𝛼.(2.25)
The function in the r.h.s. is monotone decreasing in 𝑞, so
𝑎2𝑎1≤21⋅31−𝛼≤23√9<1.(2.26)
Thus, {𝑎𝑘}∞𝑘=1 is a strictly decreasing sequence. Since all (𝑎2𝑘−1−𝑎2𝑘) are strictly positive, it follows that
∞𝑘=0(−1)𝑘𝑎𝑘𝑎=−1−𝑎2+𝑎3−𝑎4𝑎+⋯+2𝑘−1−𝑎2𝑘+⋯<0.(2.27)Case 2. 1/3≤𝛼≤1/2.Estimate (2.24) implies that ∑∞𝑘=5(−1)𝑘𝑎𝑘<0. To prove the theorem, it suffices to show that 𝑎1−𝑎2+𝑎3−𝑎4>0 when 𝑞≥2. Denoting 𝑎𝑖=(𝑞(𝑞−1)𝛼/(𝑞−1))𝑔𝑖(𝑞), 𝑖=1,2,3,4, we write the following:
𝑎1−𝑎2+𝑎3−𝑎4=𝑞(𝑞−1)𝛼𝑔𝑞−11(𝑞)−𝑔2(𝑞)+𝑔3(𝑞)−𝑔4𝑞(𝑞)=∶(𝑞−1)𝛼𝑞−1𝐾(𝑞).(2.28)
We are left to show that 𝐾(𝑞) is strictly positive for the specified values of 𝑞 and 𝛼. First of all, notice that 𝑔1(𝑞)=1, while 𝑔2(𝑞),𝑔3(𝑞), and 𝑔4(𝑞) are strictly decreasing in 𝑞 on (0,+∞). Hence, for 𝑞∈[2,5/2],
𝐾(𝑞)≥1−𝑔1(2)+𝑔252−𝑔32(2)=1−3⋅3𝛼+2002457394𝛼−8315⋅15𝛼=∶𝐿(𝛼).(2.29)
The function 𝐿(𝛼) is strictly decreasing on [1/3,1/2]. Indeed,
𝐿2(𝛼)=−3⋅3𝛼ln3+2002457394𝛼ln394−8315⋅15𝛼ln15(2.30)
and, for 𝛼∈[1/3,1/2],
𝐿2(𝛼)≤−3⋅31/3ln3+20024573941/2ln394−8315⋅151/3ln15≤−0.4332<0,(2.31)
whence 𝐿(𝛼)≥𝐿(1/2)≥1.096×10−3>0 for 𝛼∈[1/3,1/2].Similarly, for 𝑞∈[5/2,3],
𝐾(𝑞)≥1−𝑔152+𝑔2(3)−𝑔352=1−10⋅7212𝛼+9208⋅13𝛼−8000⋅14963132038𝛼=∶𝑀(𝛼).(2.32)
Applying the same reasoning as done for 𝐿(𝛼), it can be shown that 𝑀(𝛼) is strictly decreasing on [1/3,1/2]. Since 𝑀(1/2)≥0.238>0, it follows that 𝑀(𝛼)>0 for all 𝛼∈[1/3,1/2].Finally, for 𝑞∈[3,+∞), we obtain
𝐾(𝑞)≥1−𝑔1(3)−𝑔33(3)=1−8⋅4𝛼−2716640⋅40𝛼=∶𝑁(𝛼).(2.33)
Obviously, 𝑁(𝛼) is a strictly decreasing function for all 𝛼∈ℝ, whence, for 𝛼∈[1/3,1/2],
1𝑁(𝛼)≥𝑁2≥0.239>0,(2.34)
which completes the proof.

Remark 2.6. It can be seen from the proof that, the statement of the theorem is true for any 𝛼∈(0,1) and 𝑞≥𝑞0(𝛼).

An illustrative example is supplied below.

Example 2.7. Let 𝑓(𝑥)=3√𝑥. The graphs of 𝑦=𝑓(𝑥) and 𝑦=𝐵𝑛,𝑞(𝑓;𝑥) for 𝑞=2 and 𝑛=4,5 are exhibited in Figure 1. Similarly, Figure 2 represents the graphs of 𝑦=𝑓(𝑥) and 𝑦=𝐵𝑛,𝑞(𝑓;𝑥) for 𝑞=2 and 𝑛=6,7 over the subintervals [0,0.5] and [0.5,1], respectively. In addition, Table 1 presents the values of the error function 𝐸(𝑛,𝑞,𝑥)∶=𝐵𝑛,𝑞(𝑓;𝑥)−𝑓(𝑥) with 𝑞=2 at some points 𝑥∈[0,1]. The points are taken both in 𝕁𝑞 and in [0,1]⧵𝕁𝑞. It can be observed from Table 1 that, while at the points 𝑥∈𝕁𝑞, the values of the error function are close to 0, at the points 𝑥∉𝕁𝑞, the values of the error function may be very large in magnitude.

Table 1: The values of 𝐸(𝑛,𝑞,𝑥)=𝐵𝑛,𝑞(𝑓;𝑥)−𝑓(𝑥) at some points 𝑥∈[0,1].

Figure 1: Graphs of 𝑦=𝑓(𝑥) and 𝑦=𝐵𝑛,2(𝑓;𝑥), 𝑛=4,5.

Figure 2: Graphs of 𝑦=𝑓(𝑥) and 𝑦=𝐵𝑛,2(𝑓;𝑥), 𝑛=6,7.

Remark 2.8. Table 1 also shows that while the error function changes its sign for different values of 𝑥, for 𝑥=𝑞−𝑗∈𝕁𝑞, its values are negative, that is, 𝐵𝑛,𝑞(𝑡1/3;𝑞−𝑗)<𝑓(𝑞−𝑗) for 𝑞−𝑗∈𝕁𝑞. This is a particular case of the following statement.

Theorem 2.9. Let 𝑞>1. If 𝑓(𝑥) is convex (concave) on [0,1], then
𝐵𝑛,𝑞𝑓;𝑞−𝑗𝑞≥𝑓−𝑗correspondingly𝐵𝑛,𝑞𝑓;𝑞−𝑗𝑞≤𝑓−𝑗,(2.35)
for all 𝑞−𝑗∈𝕁𝑞.

Proof. It can be readily seen from (1.10) and (1.11) that
𝑛𝑘=0𝑝𝑛𝑘𝑞;𝑞−𝑗=1,𝑛𝑘=0[𝑘]𝑞[𝑛]𝑞𝑝𝑛𝑘𝑞;𝑞−𝑗=𝑞−𝑗,(2.36)
while 𝑝𝑛𝑘(𝑞;𝑞−𝑗)≥0. By virtue of Jensen's inequality, if 𝑓 is convex on [0,1], then whenever 𝑛∈ℕ and 𝑥0,𝑥1,…,𝑥𝑛∈[𝑎,𝑏], there holds the following:
𝑛𝑘=0𝜆𝑘𝑓𝑥𝑘≥𝑓𝑛𝑘=0𝜆𝑘𝑥𝑘.(2.37)
for all 𝜆0,𝜆1,…,𝜆𝑛≥0 satisfying ∑𝑛𝑘=0𝜆𝑘=1. Setting
𝑥𝑘=[𝑘]𝑞[𝑛]𝑞,𝜆𝑘=𝑝𝑛𝑘𝑞;𝑞−𝑗,𝑘=0,1,…,𝑛,(2.38)
and observing that
𝑛𝑘=0𝜆𝑘𝑓𝑥𝑘=𝐵𝑛,𝑞𝑓;𝑞−𝑗,(2.39)
the required result is derived.

Example 2.10. Let
𝑓⎧⎪⎨⎪⎩𝑞(𝑥)=2𝑥if0≤𝑥≤𝑞−2,𝑞2𝑞𝑥−2𝑥−12𝑞2−1if𝑞−2<𝑥≤1.(2.40)
The function is concave on [0,1] and, hence, according to the previous results, 𝐵𝑛,𝑞(𝑓;𝑞−𝑗)→𝑓(𝑞−𝑗) as 𝑛→∞ from below for all 𝑗∈ℤ+. To examine the behavior of polynomials 𝐵𝑛,𝑞(𝑓;𝑥) for 𝑥∉𝕁𝑞, consider the auxiliary function:
𝑔(𝑥)=𝑓(𝑥)−𝑞2⎧⎪⎨⎪⎩0𝑥=if0≤𝑥≤𝑞−2,−𝑞2𝑥−12𝑞2−1if𝑞−2<𝑥≤1.(2.41)
Since [𝑛−𝑘]𝑞/[𝑛]𝑞≤𝑞−𝑘 for 𝑘=0,1,…,𝑛, and [𝑛−1]𝑞/[𝑛]𝑞≥𝑞−2 whenever 𝑞𝑛≥𝑞+1, it follows that, for sufficiently large 𝑛,
𝐵𝑛,𝑞[](𝑔;𝑥)=𝑔𝑛−1𝑞[𝑛]𝑞𝑝𝑛,𝑛−1(𝑞;𝑥)+𝑔(1)𝑝𝑛𝑛(𝑥).(2.42)
Plain computations reveal
𝑔[]𝑛−1𝑞[𝑛]𝑞=−(𝑞−1)(𝑞𝑛)−𝑞−12(𝑞𝑛−1)2,(𝑞+1)(2.43)
yielding
𝐵𝑛,𝑞(𝑔;𝑥)=−(𝑞𝑛)−𝑞−12(𝑞𝑛𝑥−1)(𝑞+1)𝑛−1𝑞(1−𝑥)−2𝑥−1𝑛.(2.44)
Consequently, for 𝑥∉𝕁𝑞, one obtains
lim𝑛→∞𝐵𝑛,𝑞⎧⎪⎨⎪⎩0(𝑔;𝑥)=if|𝑥|<𝑞−1,∞if|𝑥|>𝑞−1.(2.45)
Since, by (1.10), 𝐵𝑛,𝑞(𝑓;𝑥)=𝑞2𝑥+𝐵𝑛,𝑞(𝑔;𝑥), it follows that:
lim𝑛→∞𝐵𝑛,𝑞⎧⎪⎪⎨⎪⎪⎩𝑞(𝑓;𝑥)=2𝑥if|𝑥|<𝑞−1,∞if|𝑥|>𝑞−1𝑞,𝑥≠1,2+1𝑞+1if𝑥=𝑞−1,1if𝑥=1.(2.46)
For 𝑥=−𝑞−1, the limit does not exist. Additionally, it is not difficult to see that 𝐵𝑛,𝑞(𝑓;𝑥)→𝑓(𝑥) as 𝑛→∞ uniformly on any compact set inside (−1/𝑞2,1/𝑞2), while on any interval outside of (−1/𝑞2,1/𝑞2), the function 𝑓(𝑥) is not approximated by its 𝑞-Bernstein polynomials. This agrees with the result from [17], Theorem 2.3. The graphs of 𝑓(𝑥) and 𝐵𝑛,𝑞(𝑓;𝑥) for 𝑞=2, 𝑛=5 and 8 on [0,1] are given in Figure 3. The values of the error function at some points 𝑥∈𝕁𝑞 and at some exemplary points 𝑥∉𝕁𝑞 are given in Table 2.

Table 2: The values of 𝐸(𝑛,𝑞,𝑥)=𝐵𝑛,𝑞(𝑓;𝑥)−𝑓(𝑥) at some points 𝑥∈[0,1].

Figure 3: Graphs of 𝑦=𝑓(𝑥) and𝑦=𝐵𝑛,2(𝑓;𝑥), 𝑛=5,8.

Remark 2.11. Following Charalambides [9], consider a sequence of random variables {𝑋𝑛(𝑗)}∞𝑛=1 possessing the distributions 𝑃𝑛(𝑗) given by
𝐏𝑋𝑛(𝑗)=[]𝑛−𝑘𝑞[𝑛]𝑞=𝑝𝑛,𝑛−𝑘𝑞;𝑞−𝑗,𝑘=0,1,…,𝑛.(2.47)
Let 𝐼(𝑞−𝑗) denote a random variable with the 𝛿-distribution concentrated at 𝑞−𝑗. Theorem 2.1 implies that 𝑋𝑛(𝑗)→𝐼(𝑞−𝑗) in distribution.

Generally speaking, Theorem 2.1 shows that the 𝑞-Bernstein polynomials with 𝑞>1 possess an “interpolation-type” property on 𝕁𝑞. Information on interpolation of functions with nodes on a geometric progression can be found in, for example, [18] by Schoenberg.

3. On the 𝑞-Bernstein Polynomials of the Weierstrass-Type Functions

In this section, the 𝑞-Bernstein polynomials of the functions with “bad” smoothness are considered. Let 𝜑(𝑥)∈𝐶[−1,1] satisfy the condition:
𝜑(0)>𝜑(𝑥)for[]𝑥∈−1,1⧵{0}.(3.1)
The letter 𝜑 will also denote a 2-periodic continuation of 𝜑(𝑥) on (−∞,∞).

Definition 3.1. Let 𝑎,𝑏∈ℝ satisfy 0<𝑎<1<𝑎𝑏. A function 𝑓(𝑥) is said to be Weierstrass-type if
𝑓(𝑥)=∞𝑘=0𝑎𝑘𝜑𝑏𝑘𝑥.(3.2)
Notice that 𝑓(𝑥) is continuous if and only if 𝜑(−1)=𝜑(1). For 𝜑(𝑥)=cos𝜋𝑥 and a special choice of 𝑎 and 𝑏 (see, e.g., [19, Section 4]), the classical Weierstrass continuous nowhere differentiable function is obtained. In [19], one can also find an exhaustive bibliography on this function and similar ones. For 𝜑(𝑥)=1−|𝑥|, a function analogous to the Van der Waerden continuous nowhere differentiable function appears.

The aim of this section is to prove the following statement.

Theorem 3.2. If 𝑓(𝑥) is a Weierstrass-type function, then the sequence 𝐵𝑛,𝑞(𝑓;𝑥) of its 𝑞-Bernstein polynomials is not uniformly bounded on any interval [0,𝑐].

Proof. To prove the theorem, the following representation of 𝑞-Bernstein polynomials (see [15], formulae (6) and (7)) is used:
𝐵𝑛,𝑞(𝑓;𝑥)=𝑛𝑘=0𝜆𝑘𝑛𝑓10;[𝑛]𝑞[𝑘];…;𝑞[𝑛]𝑞𝑥𝑘,(3.3)
where
𝜆0𝑛=𝜆1𝑛=1,𝜆𝑘𝑛=11−[𝑛]𝑞⋯[]1−𝑘−1𝑞[𝑛]𝑞,𝑘=2,…,𝑛,(3.4)
and 𝑓[𝑥0;𝑥1;…;𝑥𝑘] denote the divided differences of 𝑓, that is,
𝑓𝑥0𝑥=𝑓0𝑥,𝑓0;𝑥1=𝑓𝑥1𝑥−𝑓0𝑥1−𝑥0𝑓𝑥,…,0;𝑥1;…;𝑥𝑘=𝑓𝑥1;…;𝑥𝑘𝑥−𝑓0;…;𝑥𝑘−1𝑥𝑘−𝑥0.(3.5)
When 𝑞=1, the well-known representation for the classical Bernstein polynomials is recovered and the numbers 𝜆𝑘𝑛 are the eigenvalues of the Bernstein operator, see [20], Chapter 4, Section 4.1 and [21]. The latter result has been extended to the case 𝑞≠1 in [15]. Clearly, it suffices to consider the case 0<𝑐<1. From (3.3), it follows that
𝐵𝑛,𝑞(𝑓;0)=𝜆1𝑛𝑓10;[𝑛]𝑞=[𝑛]𝑞𝑓1[𝑛]𝑞,−𝑓(0)(3.6)
and, hence,
||𝐵𝑛,𝑞(||=[𝑛]𝑓;0)𝑞1𝑓(0)−𝑓[𝑛]𝑞=[𝑛]𝑞∞𝑘=0𝑎𝑘𝑏𝜑(0)−𝜑𝑘[𝑛]𝑞.(3.7)
What remains is to find a lower bound for |𝐵𝑛,𝑞(𝑓;0)|. Due to (3.1), all terms of the series are nonnegative and, therefore,
||𝐵𝑛,𝑞||≥[𝑛](𝑓;0)𝑞𝑎𝑗𝑏𝜑(0)−𝜑𝑗[𝑛]𝑞forany𝑗=0,1,…(3.8)
Let 𝑗=𝑗𝑛 be chosen in such a way that
1𝑏<𝑏𝑗𝑛[𝑛]𝑞≤1.(3.9)
For 𝑛>𝑏, such a choice is possible because, in this case, inequality (3.9) implies that
[𝑛]0<ln𝑞ln𝑏−1<𝑗𝑛≤[𝑛]ln𝑞.ln𝑏(3.10)
Since the length of the interval (ln[𝑛]𝑞/ln𝑏−1,ln[𝑛]𝑞/ln𝑏] is 1, there is a positive integer, say, 𝑗𝑛, such that 𝑗𝑛∈(ln[𝑛]𝑞/ln𝑏−1,ln[𝑛]𝑞/ln𝑏]. The obvious inequality [𝑛]𝑞>𝑞𝑛−1 implies the following:
𝑗𝑛≥(𝑛−1)ln𝑞ln𝑏−1=∶𝑛ln𝑞ln𝑏−𝐴,(3.11)
with 𝐴=(ln𝑞/ln𝑏)+1 being a positive constant. Then, for 𝑛>𝑏, it follows that
||𝐵𝑛,𝑞||≥[𝑛](𝑓;0)𝑞𝑎𝑗𝑛min[]𝑡∈1/𝑏,1[𝑛]{𝜑(0)−𝜑(𝑡)}∶=𝜏𝑞𝑎𝑗𝑛,(3.12)
where 𝜏>0 due to (3.1). Consequently,
||𝐵𝑛,𝑞(||[𝑛]𝑓;0)≥𝜏𝑞𝑏𝑗𝑛(𝑎𝑏)𝑗𝑛≥𝜏(𝑎𝑏)𝑗𝑛≥𝜏(𝑎𝑏)𝑛(lnq/ln𝑏)−𝐴,(3.13)
which leads to
||𝐵𝑛,𝑞(||𝑓;0)≥𝐶𝜌𝑛,(3.14)
where 𝐶=𝜏(𝑎𝑏)−𝐴 is a positive constant and 𝜌=(𝑎𝑏)(lnq/ln𝑏)>1. Now, assume that {𝐵𝑛,𝑞(𝑓;𝑥)} is uniformly bounded on [0,𝑐], that is, |𝐵𝑛,𝑞(𝑓;𝑥)|≤𝑀 for all 𝑥∈[0,𝑐]. By Markov's Inequality (cf., e.g., [22], Chapter 4, Section 1, pp. 97-98) it follows that
||𝐵𝑛,𝑞||≤(𝑓;0)2𝑀𝑐𝑛2∀𝑛=1,2,…,(3.15)
This proves the theorem because the latter estimate contradicts (3.14).

To present an illustrative example, let us denote the 𝑁th partial sum of the series in (3.2) by ℎ𝑁, that is:
ℎ𝑁(𝑥)=𝑁𝑘=0𝑎𝑘𝜑𝑏𝑘𝑥.(3.16)
Clearly, the function ℎ𝑁 is an approximation of (3.2) satisfying the error estimate
𝐸𝑁||(𝑥)=𝑓(𝑥)−ℎ𝑁||(𝑥)≤max[]𝑡∈−1,1||||𝑎𝜑(𝑡)𝑁+1[].1−𝑎,∀𝑥∈0,1(3.17)

Example 3.3. Let 𝜑(𝑥)=(cos𝜋𝑥),𝑎=1/2, and 𝑏=4. For 𝑁=20, one has 𝐸20(𝑥)≤10−6. The graphs of ℎ20(𝑥) and the associated 𝑞-Bernstein polynomials 𝐵𝑛,𝑞(ℎ20;𝑥) for 𝑞=2, 𝑛=4,5, and 6 on the subintervals [0,0.55] and [0.55,1] are presented in Figures 4 and 5, respectively.

Figure 4: Graphs of 𝑦=ℎ20(𝑥) and 𝑦=𝐵𝑛,2(𝑓;𝑥), 𝑛=4,5,6.

Figure 5: Graphs of 𝑦=ℎ20(𝑥) and 𝑦=𝐵𝑛,2(𝑓;𝑥), 𝑛=4,5,6.

Acknowledgment

The authors would like to express their sincere gratitude to Mr. P. Danesh from Atilim University Academic Writing and Advisory Centre for his help in the preparation of the paper.